On products of skew rotations

Let $\{S_1^t\},\ldots,\{S_n^t\}$ be the one-parametric groups of shifts along the orbits of Hamiltonian systems generated by time-independent Hamiltonians $H_1,\ldots, H_n$ with one degree of freedom. In some problems of population genetics there appear planar transformations having the form $S^{h_n}_n\cdots S_1^{h_1}$ under some conditions on Hamiltonians $H_1,\ldots,H_n$. In this paper we study asymptotical properties of trajectories of such transformations. We show that under classical non-degeneracy condition on the Hamiltonians the trajectories stay in the invariant annuli for generic combinations of lengths $h_1,\dots, h_n$, while for the special case $h_1+\dots+h_n=0$ there exists a trajectory escaping to infinity.